Simple Nature - Light and Matter

energy in the electrical energy due to its x displacement along the xaxis, and equal amounts for y and z. This is known as equipartition,meaning equal partitioning, or equal sharing. The equipartition theoremsays that if the expression for the energy looks like a sum ofsquared variables, then each degree of freedom has an average energyof 1 2kT . Thus, very generally, we can interpret temperature asthe average energy per degree of freedom (times k/2).An unexpected glimpse of the microcosmYou may have the feeling at this point that of course Boltzmannwas right about the literal existence of atoms, but only very sophisticatedexperiments could vindicate him definitively. After all,the microscopic and macroscopic definitions of entropy are equivalent,so it might seem as though there was no real advantage to themicroscopic approach. Surprisingly, very simple experiments are capableof revealing a picture of the microscopic world, and there isno possible macroscopic explanation for their results.In 1819, before Boltzmann was born, Clément and Desormes didan experiment like the one shown in figure j. The gas in the flaskis pressurized using the syringe. This heats it slightly, so it is thenallowed to cool back down to room temperature. Its pressure ismeasured using the manometer. The stopper on the flask is poppedand then immediately reinserted. Its pressure is now equalized withthat in the room, and the gas’s expansion has cooled it a little,because it did mechanical work on its way out of the flask, causingit to lose some of its internal energy E. The expansion is carried outquickly enough so that there is not enough time for any significantamount of heat to flow in through the walls of the flask before thestopper is reinserted. The gas is now allowed to come back upto room temperature (which takes a much longer time), and as aresult regains a fraction b of its original overpressure. During thisconstant-volume reheating, we have P V = nkT , so the amount ofpressure regained is a direct indication of how much the gas cooleddown when it lost an amount of energy ∆E.j / An experiment for determiningthe shapes of molecules.k / The differing shapes of ahelium atom (1), a nitrogenmolecule (2), and a difluoroethanemolecule (3) havesurprising macroscopic effects.Section 5.4 Entropy As a Microscopic Quantity 323